Abstract

Our aim in this article is to incorporate the notion of “strongly s-convex function” and prove a new integral identity. Some new inequalities of Simpson type for strongly s-convex function utilizing integral identity and Holder’s inequality are considered.

Keywords

Mathematics Subject Classification

26D1526E6041A55

Background

The following definition is well known in the literature as convex function:

Let \(f:I\subseteq R\rightarrow R\) be a function defined on the interval I of real numbers. Then f is called convex, if \(f\left( {\lambda x+\left( {1-\lambda } \right) y} \right) \le \lambda f\left( x \right) +\left( {1-\lambda } \right) f\left( y \right) ,\) for all \(x,y\in I\) and \(\lambda \in \left[ {0,1} \right] .\) Geometrically, this means that if P, Q and R are three distinct points on graph of f with Q between P and R, then Q is on or below chord PR.

where the mapping \(f:[a,b]\rightarrow R\) is supposed to be a four times continuously differentiable on the interval \(\left( {a,b} \right)\) and having the fourth derivative bounded on \(\left( {a,b} \right) ,\) that is

Dragomir et al. (2000) proved that: Let \(f:[a, b] \rightarrow R\) be a differentiable function on \(I^0\) (interior of I) \(a,b\in I\) with \(a<b.\) If the mapping \(\left| {{f}'} \right|\) is convex on \(\left[ {a,b} \right] ,\) then we have the following inequality:

Sarikaya et al. (2010) showed that: Let \(f: [a, b]\rightarrow R\) be a differentiable function on \(I^0\)(interior of I) such that \(f^{'} \in L_{1}[a, b]\), where \(a,b\in I\) with \(a<b.\) If the mapping \(\left| {{f}'} \right|\) is s-convex on \(\left[ {a,b} \right] ,\) for some fixed \(s\in \left( {0,1} \right] ,\) then we have the following inequality

Alomari et al. (2011) established that: Let \(f:I\subset R\rightarrow R\) be a differentiable function on \(I^0\) (interior of I)\(a,b\in I\) with \(a<b.\) If the mapping \(\left| {{f}'} \right|\) is s-convex on \(\left[ {a,b} \right] ,\) for some \(s\in \left( {0,1} \right] ,\) then we have the following inequality:

Theorem 6

Letf be defined as in Theorem4 and the mapping \(\left| {{f}''} \right| ^q\)is strongly s-convex on\(\left[ {a,b} \right] ,\)for\(q>1\)and for some fixed\(s\in \left( {0,1} \right]\), then we have the following inequality:

Theorem 7

Let f be defined as in Theorem 4and the mapping \(\left| {{f}''} \right| ^q\)is strongly s-convex on\(\left[ {a,b} \right] ,\)for\(q>1\)and for some fixed\(s\in \left( {0,1} \right]\), then we have the following inequality:

Theorem 8

Letfbe defined as in Theorem4and the mapping \(\left| {{f}''} \right| ^q\)is strongly s-convex on \(\left[ {a,b} \right] ,\)for\(q>1\)and for some fixed\(s\in \left( {0,1} \right]\), then we have the following inequality:

Conclusion

We incorporated notion of “strongly s-convex function” and proved a new integral identity. Some new inequalities of Simpson type for strongly s-convex function utilizing integral identity and Holder’s inequality are also considered. These results give better estimates as presented earlier in the literature.

Declarations

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

There authors are thankful to the reviewers for their fruitful comments towards the improvement of the paper.

Competing interest

The authors declare that they have no competing interests.

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Wang Y, Wang SH, Qi F (2013) Simpson type integral inequalities in which the power of the absolute value of the first derivative of the integrand is s-preinvex. Facta Univ Ser Math Inform 28(2):151–159Google Scholar